WASHINGTON — Gay rights groups are pushing to adjust a bipartisan Senate bill to include gay couples, but Democrats are treading carefully, wary of adding another divisive issue that could lose Republican support and jeopardize the entire bill.

Both parties want the bill to succeed. Merely getting to agreement on the basic framework for the immigration overhaul, which would create a long and costly path to citizenship for the estimated 11 million people in the U.S. illegally, was no small feat for senators.

And getting it through a divided Congress is still far from a done deal.

Even so, gay rights groups, their lobbyists and grass-roots supporters are insisting the deal shouldn’t exclude bi-national, same-sex couples – about 28,500 of them, according to a 2011 study from the Williams Institute at UCLA Law.

They’re ramping up a campaign to change the bill to allow gay Americans to sponsor their partners for green cards, the same way straight Americans can. Supporters trekked to the Capitol to make their case at senators’ offices on Wednesday.

“Opponents will be proposing amendments that, if passed, could collapse this very fragile coalition that we’ve been able to achieve,” Sen. John McCain, an Arizona Republican, said last week at the unveiling of the bill. He said the eight senators from both parties who crafted the legislation are committed to voting against changes that could kill it.

For Democrats, it’s a precarious position to be in. Democratic senators overwhelmingly support gay marriage – all but three are now on the record voicing their support – and two dozen of them this year backed a separate bill called the Uniting American Families Act to let gays sponsor their partners independent of a comprehensive immigration overhaul.

But the party’s senators are still bruised from an agonizing defeat on gun control this month. And few seem eager to inject divisive issues that might sink their best prospects for a major legislative victory this year and a potential keystone of President Barack Obama’s legacy.

Support from both Hispanics and gays was critical to Obama’s re-election, and his overwhelming advantage among Hispanics was a major factor prompting Republicans to warm to immigration overhaul almost immediately after. But now, one community’s gain on the immigration front could be to the other’s detriment.

Democratic Sen. Patrick Leahy, who chairs the Senate Judiciary Committee, has committed to offering an amendment to the bill to allow gay citizens to sponsor their partners, said Ty Cobb, an attorney and lobbyist with the Human Rights Campaign, a gay rights group.

Another Democratic senator, Al Franken, pledged in a Judiciary hearing on the bill Monday to do “everything we can” to adjust the bill.

But even if the amendment makes it through the Senate, it faces a tougher path if and when the bill moves to the Republican-controlled House of Representatives.

Republican leaders there have been defending the Defense of Marriage Act, which defines marriage as between a man and a woman, though Obama has said it is unconstitutional. And while Obama supports same-sex marriage, his administration has shown little appetite for forcing the issue while the immigration overhaul’s prospects are still shaky.

“No one will get everything they want from it, including the president. That’s the nature of compromise. But the bill is largely consistent with the principles he has laid out repeatedly,” Obama spokesman Jay Carney said last week. A White House spokesman declined to answer further questions about the issue.

Some Democrats argue privately that with the Supreme Court poised to rule on the constitutionality of the Defense of Marriage Act, which prohibits the government from giving federal marriage benefits to gay couples, the issue could soon be moot.

Still, even if the high court strikes the law down, it wo uld only bring partial relief; only couples married in the nine states that recognize gay marriages would probably be eligible.

The issue has generated an intense advocacy campaign, with gay rights organizations and Hispanic groups such as the National Council of La Raza squaring off with religious interests such as the U.S. Conference of Catholic Bishops, which sent a letter to Obama telling him including the provision could jeopardize the whole bill.

At the Human Rights Campaign, four of its seven federal lobbyists are engaged in pushing lawmakers to back such an amendment. Immigration Equality, another group supporting the provision, said it was bringing more than 60 families from 24 states to the Capitol on Wednesday to ask lawmakers to offer their support.

And Log Cabin Republicans, a gay conservative group, is making a pro-business pitch with potential Republican supporters, arguing that including gay couples would allow U.S. companies to retain the b est talent instead of forcing good workers to leave the U.S. to be with their partners.